Evaluate and simplify y'.
y = (x²+1)³ / (x⁴+7)⁸(2x+1)⁷

검증된 단계별 안내

First, identify the function y in the equation y'.y = (x²+1)³ / (x⁴+7)⁸(2x+1)⁷. Here, y is implicitly defined, and y' represents the derivative of y with respect to x.

To find y', use implicit differentiation. Differentiate both sides of the equation with respect to x. Remember that y is a function of x, so when differentiating y, apply the chain rule: d(y)/dx = y'.

Differentiate the right-hand side of the equation using the quotient rule. The quotient rule states that if you have a function f(x)/g(x), its derivative is (f'(x)g(x) - f(x)g'(x)) / (g(x))².

Apply the chain rule to differentiate the terms (x²+1)³, (x⁴+7)⁸, and (2x+1)⁷. For example, the derivative of (x²+1)³ is 3(x²+1)² * 2x, using the chain rule.

After differentiating, simplify the expression by combining like terms and factoring where possible. This will give you the expression for y' in terms of x.

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주요 개념

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Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to its variable. In this context, we need to apply differentiation rules to the given function y, which is a quotient of two complex expressions. Understanding how to differentiate polynomial and composite functions is essential for solving the problem.

Quotient Rule

The Quotient Rule is a formula used to differentiate functions that are expressed as the ratio of two other functions. It states that if y = u/v, then y' = (u'v - uv')/v², where u and v are functions of x. This rule is crucial for this problem since y is a quotient, and applying it correctly will allow us to find y'.

Chain Rule

The Chain Rule is a fundamental technique in calculus for differentiating composite functions. It states that if a function y is composed of another function u, then the derivative y' can be found by multiplying the derivative of u with respect to x by the derivative of y with respect to u. This concept is important here as the function y involves powers of polynomials, requiring the Chain Rule for proper differentiation.

Evaluate and simplify y'.y = (x²+1)³ / (x⁴+7)⁸(2x+1)⁷